Theorem maprnin 28894

Description: Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.)

Hypotheses

Ref	Expression
maprnin.1	⊢ 𝐴 ∈ V
maprnin.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
maprnin	⊢ ((𝐵 ∩ 𝐶) ↑𝑚 𝐴) = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ran 𝑓 ⊆ 𝐶}

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓

Proof of Theorem maprnin

Step	Hyp	Ref	Expression
1		ffn 5958	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
2		df-f 5808	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐶 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐶))
3	2	baibr 943	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → (ran 𝑓 ⊆ 𝐶 ↔ 𝑓:𝐴⟶𝐶))
4	1, 3	syl 17	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → (ran 𝑓 ⊆ 𝐶 ↔ 𝑓:𝐴⟶𝐶))
5	4	pm5.32i 667	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ran 𝑓 ⊆ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐶))
6		maprnin.2	. . . . . 6 ⊢ 𝐵 ∈ V
7		maprnin.1	. . . . . 6 ⊢ 𝐴 ∈ V
8	6, 7	elmap 7772	. . . . 5 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝐵)
9	8	anbi1i 727	. . . 4 ⊢ ((𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ ran 𝑓 ⊆ 𝐶))
10		fin 5998	. . . 4 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐶))
11	5, 9, 10	3bitr4ri 292	. . 3 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶))
12	11	abbii 2726	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
13	6	inex1 4727	. . 3 ⊢ (𝐵 ∩ 𝐶) ∈ V
14	13, 7	mapval 7756	. 2 ⊢ ((𝐵 ∩ 𝐶) ↑𝑚 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)}
15		df-rab 2905	. 2 ⊢ {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ran 𝑓 ⊆ 𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
16	12, 14, 15	3eqtr4i 2642	1 ⊢ ((𝐵 ∩ 𝐶) ↑𝑚 𝐴) = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ran 𝑓 ⊆ 𝐶}

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ran crn 5039   Fn wfn 5799  ⟶wf 5800  (class class class)co 6549   ↑_𝑚 cmap 7744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746

This theorem is referenced by:  fpwrelmapffs  28897